Rapid expansion in finite simple groups

TL;DR

This paper demonstrates that small normal subsets in finite simple groups expand rapidly, with their product sets growing almost quadratically in size, highlighting significant expansion properties in these groups.

Contribution

The paper establishes a near-quadratic growth rate for the product of small normal subsets in finite simple groups, revealing new expansion phenomena.

Findings

Small normal subsets expand rapidly in finite simple groups.

Product sets of these subsets grow almost quadratically.

The expansion rate can be made arbitrarily close to quadratic.

Abstract

We show that small normal subsets $A$ of finite simple groups expand very rapidly -- namely, $|A^2| \ge |A|^{2-\epsilon}$, where $\epsilon >0$ is arbitrarily small.